In this paper, we investigate a coverage extension scheme based on orthogonal random precoding (ORP) for the downlink of massive multiple-input multiple-output (MIMO) systems. In this scheme, a precoding matrix consisting of orthogonal vectors is employed at the transmitter to enhance the maximum signal-to-interference-plus-noise ratio (SINR) of the user. To analyze and optimize the ORP scheme in terms of cell coverage, we derive the analytical expressions of the downlink coverage probability for two receiver structures, namely, the single-antenna (SA) receiver and multiple-antenna receiver with antenna selection (AS). The simulation results show that the analytical expressions accurately capture the coverage behaviors of the systems employing the ORP scheme. It is also shown that the optimal coverage performance is achieved when a single precoding vector is used under the condition that the threshold of the signal-to-noise ratio of the coverage is greater than one. The performance of the ORP scheme is further analyzed when different random precoder groups are utilized over multiple time slots to exploit precoding diversity. The numerical results show that the proposed ORP scheme over multiple time slots provides a substantial coverage gain over the space-time coding scheme despite its low feedback overhead.